Being the leader of a rescue force in this game, your major role will be to set out on missions to help other Pokémon. Your first mission actually takes place before even your rescue force is formed, and that is when you and your partner set out to help find Butterfree's lost Caterpie child in the Small Forest. After completing this first mission, your partner will suggest that you two form an official rescue force, which seems to be a very popular job in the Pokémon world in the game. You get to pick the name of your team, and following your second compulsory mission to save the Magnemite at the Thunderwave Cave, you are formerly ready to receive missions on your own.
Whenever you visit a dungeon, you are already considered to be on a mission, because if you happen to lose, the consequence will be the same as when you actually go on a rescue mission. Your money and items will be halved or completely gone if this happens, so deposit them at the Persian Bank and Kangaskhan Warehouse for safety measures unless you really need that item in the dungeon.
Otherwise, there are two major type of missions that you accept in this game. The first kind are compulsory ones directly related to the game plot, which unlocks new dungeons to explore. Most of these missions will come to you automatically as the game progresses, so a majority of them you don't have to look for. There are several exceptions however, which you need to find at the notice board outside the Pelipper Contact House to the right of the Pokémon Sqaure. See below for more details.

The second type of missions are optional ones which you do to get rare items or simply to raise the rank of your rescue force. For these kind of missions, there are 7 categories of difficulty, E, D, C, B, A, S or *, which are marked at the end of each mission request letter or post. You find these either at your mail box outside your headquarters (brought to you by a Pelipper), or from the notice borad outside the Pelipper Contact House.
The E category ones are the easiest to complete and usually involve the earlier dungeons with lesser floors. They almost always are simple rescue missions where you find a lost Pokémon, but sometimes it is the lost Pokémon's friend who posted the request, so it may join you on the mission and you must work to keep it alive too. You are rewarded with 5 rank points, and the prize is usually Pokedollars or some common items such as seeds or berries. The D (10 to 20 points), C (40 to 60 points), B (80 to 100 points) and A (150 to 200 points) category ones are progressively more difficult, and at times the mission will request an item (it will be mentioned in the letter or post) instead of a simple rescue. The rewards are usually better such as TMs, rare seeds and accessories.
The S (300 to 400) and * (700 to 1500) category missions are considerably more difficult, in the sense that they usually involve the dungeons with more floors, and your target is almost always on floors higher than 50. Again, the mission could be of rescue nature or item request, and the prizes will be logically better. In these two categories, at times there will be missions which reward you with the Manene or Manyula statues which are placed outside your headquarters. There is known to have a pool of different missions which might reward you with these two rare items, but once you have obtained the statue, the rest will not appear so you only get one of each of these statues, and not two Manene statues or so. These statue missions are reported to be found only on the notice boards, so don't just wait for a Pelipper to drop it at your headquarters.

In general terms, the items you can receive from these optional missions include Pokedollars, TMs, berries, seeds, gummies, accessories, and on rare occasions, the Manene and Manyula statues.
So, how do you accept and go on missions afterall? At both your headquarter's mail box and the Pelipper Contact House notice board, whenever you click on a mission, you have two choices. The first is to accept it, and the second is to read more about it

When you have accepted a mission, it will appear on your Mission List menu. Go into that menu, click on the mission, and you could either deal with it now, delete it, or read about it again

The mission will only be active once you've chosen to deal with it now, and there should be a red mark on its envelope.
Now, access the dungeons by heading south from your headquarters. The dungeon where your mission takes place will have that red marked envelope next to its name.

You can do more than one mission in the same dungeon with one single visit. Once you're in the dungeons, from the menu, you could click Others (fourth option), then Current Mission (fourth option again), to remind yourself of what you're doing. When you've arrived on the right floor, a message will also remind you that you have reached your destination.
It's not too difficult to locate your target, because it is usually a Pokémon specie not found in that dungeon, and it will simply wander around and not attack you when you go near it

Talk to it (sometimes you need to change the direction you face by the 'start' method, because for some reason your target moves when you move, and it's hard to face it directly) and it will teleport away, unless it's an item request mission and you don't have the item with you. You are then asked if you wish to leave the dungeon, click yes, and you will be asked for confirmation, click yes again and you will be teloported away too. But if you click no (requires a second yes confirmation when it asks if you wish to stay), you will remain and can complete the other missions in the dungeon that you're on.

There are 2 types of such optional missions which are really not that optional afterall. They reward you with Friend Areas or unlock new dungeons which are important to the game plot.

Friend Areas:
There are 4 Friend Areas which can only be obtained from missions, namely the Blue Sky Meadow, the Rock Cave, the Moon Gaze Mountain, and the Dragon Cave. These are all in the D category and are randomly posted on the notice board or delivered to your mail box. You will not be able to befriend the Pokémon inhabiting in these Friend Areas until you have gotten them (especially Jirachi who lives at the Moon Gaze Mountain), but more importantly, the Blue Sky Meadow is required to unlock some later dungeons such as the Wish Cave and Pure Forest.

Like the statue missions, more than one mission leads to these Friend Areas, but once you have the area, the other missions which do will not appear.

Dungeons:
Uproar Forest - after you returned from the Ninetales mission at the Snow Mountain Top, do a few optional missions, and you will find a help request by a Wynaut at the Pelipper Contact House notice board. If you don't read Japanese, you will still know that you have found the right mission because once you accept it, a Wynaut and Wobbuffet will appear to grant you access to the Uproar Forest.

Howl Forest - this appears to come up randomly at the Pelipper Contact House or at your mail box after you have gotten the Blue Sky Meadow Friend Area. It will be a rescue request from a Smeargle, and once you've accepted it, there will also be a short animation between you and your partner, allowing you to access the Howl Forest.